Exercise 2 – Animated Solutions വശങ്ങളും കോണുകളും
Three questions that use Key idea 3 (all sides scaled by the same factor ⇒ same angles) and Key idea 4 (area scales by the square of the factor). Try each question in your notebook first, then press Show solution and play the animation.
- Key idea 3: if every side of a triangle is multiplied by the same number, the new triangle has the same angles. So you can copy a triangle at any size using only a ruler and compass – no protractor.
- Key idea 4: if the sides are multiplied by r, the area is multiplied by r2.
- Reminder: this "sides fix the angles" rule is special to triangles.
Question 1 · Draw a 1¼-size copy
Question: Draw a triangle with the same angles as the triangle below (sides 4 cm, 6 cm, 8 cm) and with sides 1¼ times as long.
Plan: 1¼ × 4 = 5 cm, 1¼ × 6 = 7.5 cm, 1¼ × 8 = 10 cm. Draw a triangle with these three sides using a compass. Because all three sides are scaled by the same factor, Key idea 3 guarantees the angles are the same – we never need to measure an angle.
Compare with Exercise 1 Q1, where we knew angles and copied them with a protractor. Here we know sides, so we copy them with a compass. Both methods give a triangle of the same shape – that is what "similar" means.
Question 2 · A quadrilateral scaled 1½ times
Question: The quadrilateral below has sides 4 cm, 2 cm, 3 cm, 6 cm and the diagonal shown is 5 cm.
(i) Draw a quadrilateral with the same angles and all sides 1½ times as long.
(ii) Draw a quadrilateral with different angles but with sides 1½ times these.
(i) The diagonal splits the quadrilateral into two triangles: (6, 3, 5) and (5, 2, 4). Scale each triangle by 1½: (9, 4.5, 7.5) and (7.5, 3, 6). Draw them on a common diagonal of 7.5 cm. Each triangle keeps its angles (Key idea 3), and the angles of the quadrilateral are sums of triangle angles, so the quadrilateral has the same angles.
(ii) Keep the four sides 9, 4.5, 3, 6 but choose a different diagonal, e.g. 8 cm. The triangles (9, 4.5, 8) and (8, 3, 6) both exist, so the quadrilateral exists – with the same sides and different angles. A four-sided frame is not rigid; a triangle is.
The four sides stay 9, 4.5, 3, 6 cm (exactly 1½ times the original). Only when the diagonal is 7.5 cm (1½ × 5) do the angles match the original. Any other diagonal gives a quadrilateral with the same sides but different angles – that answers part (ii).
For triangles, sides in the same ratio ⇒ same angles. For quadrilaterals (and any polygon with more sides) this is false: you also need the angles, or extra lengths like a diagonal. Remember the square and the rectangle from Section 1 – the other way round fails too.
Question 3 · How the area changes
Question: The area of a triangle is 6 sq cm. What is the area of a triangle whose sides are all 4 times as long? What if every side is halved?
Sides × 4 ⇒ area × 42 = 16 ⇒ 16 × 6 = 96 sq cm.
Sides × ½ ⇒ area × (½)2 = ¼ ⇒ 6 ÷ 4 = 1.5 sq cm.
പരപ്പളവ് മാറുന്ന തോത്, വശങ്ങൾ മാറുന്ന തോതിന്റെ വർഗമാണ്.
The scale factor of the area is the square of the scale factor of the sides: sides × r ⇒ area × r2.
A pizza of double the diameter has four times the area (so it should cost about four times as much, not twice!). A map at half the scale needs only a quarter of the paper. Enlarging a photo to 3× its width uses 9× the ink.
Check yourself
- To copy a triangle at a new size, multiply the three sides and draw with a compass – the angles take care of themselves.
- A quadrilateral must be built from triangles (use a diagonal). Scaling only the four sides does not fix its angles.
- Area changes by the square of the side scale factor: × 4 sides ⇒ × 16 area; × ½ sides ⇒ × ¼ area.
Next: Section 3 – The Third Way (two sides and the angle between them) ▶